The ideal class group is an important object in the study of algebraic number fields, as is the associated class number. In cyclotomic fields in particular, the class number decomposes into two factors. While the second factor is difficult to determine due to the need for a basis of the group of units, the first factor is given by a surprisingly simple formula. In terms of the structure of the ideal class group, Stickelberger's theorem gives an important result. The theorem provides explicit annihilators of the ideal class group, from which the Stickelberger ideal arises. In this work, the analytic class number formula is derived and related to the minus part of the index of the Stickelberger ideal in the integral group ring over the Galois group.